Joshua Zahl

Associate Professor

Research Interests

Harmonic Analysis
Discrete and Combinatorial Geometry

Relevant Degree Programs



Master's students
Doctoral students
Postdoctoral Fellows

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Combinatorial problems for intervals, fractal sets and tubes (2021)

We discuss three problems related to combinatorial geometry, based on two separate works. The first work concerns arrangements of intervals in R² for which there are many pairs forming trapezoids, meaning the convex hull of the pair is a trapezoid. We characterise arrangements forming more than a certain number of trapezoids, showing that all such sets have underlying algebraic structure. An important role is played in particular by conic curves. The proof uses a transformation from intervals in the plane to lines in R³ and then relies on a theorem of Guth and Katz on intersecting lines in R³. The second work concerns combinatorial problems for discretised sets, where objects are only distinguishable up to some small scale. Discretised sets can be used to approximate fractal sets, and our results imply improved quantitative bounds for the 1/2-Furstenberg set problem in R² and the upper Minkowski dimension of Besicovitch sets in R³, as well as slight generalisations of each of these problems. The techniques involved in this second work are mostly combinatorial and our main ingredient is the discretised sum-product theorem from additive combinatorics. In particular, we reduce the 1/2-Furstenberg set problem to the discretised-sum product problem and reduce the Besicovitch set problem to the Furstenberg set problem.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Kakeya maximal function conjecture for semialgebraic mappings (2021)

The Kakeya maximal function conjecture is a quantitative, single scale formulation of the Kakeya conjecture. Recently, algebraic methods have been leading to progress in the Kakeya family of problems. In 2018, Katz and Rogers proved a conjecture concerning the number of ?-tubes with ?-separated directions which intersect a semialgebraic set with proportion at least λ. We will discuss the proof of this result which involves real algebraic geometry. We will then use this result to prove the Kakeya maximal function conjecture for the special case when the mappings are semialgebraic.

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Cartesian products avoiding patterns (2020)

The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns such as finding a set whose Cartesian product avoids the zero set of a given function. Previous work on the subject has considered patterns described by polynomials, or functions satisfying certain regularity conditions. We provide an exposition of some results in this setting, as well as consideringnew strategies to avoid ‘rough patterns’. There are several problems that fit intothe framework of rough pattern avoidance. For instance, we prove that for any set X with lower Minkowski dimension s, there exists a set Y with Hausdorff dimension 1 − s such that for any rational numbers a₁, ..., aN, a₁Y + ··· + aNY is disjoint from X, or intersects solely at the origin. As a second application, we construct subsets of Lipschitz curves with dimension 1/2 not containing the vertices of any isosceles triangle.

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